{"pk":64915,"title":"The combinatorics of a tree-like functional equation for connected chord diagrams","subtitle":null,"abstract":"We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord diagrams leads to new enumerative insights through the study of novel diagram classes. We present a recursive bijection between connected top-cycle-free diagrams with \\(n\\) chords and triangulations of a disk with \\(n+1\\) vertices, thereby counting the former. This connects to combinatorial maps, Catalan intervals, and uniquely sorted permutations, leading to new conjectured bijective relationships between diagram classes defined by forbidding graphical subdiagrams and imposing connectedness properties and a rich variety of other combinatorial objects. We conclude by exhibiting and studying a direct bijection between diagrams of size \\(n\\) with a single terminal chord and diagrams of size \\(n-1\\).\n \nMathematics Subject Classifications: 05A05, 05A10, 05A15, 05A18, 05A19\n \nKeywords: Chord diagrams, perfect matchings, combinatorial classes, pattern avoidance, combinatorial maps, triangulations, Catalan posets, uniquely sorted permutations","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Chord diagrams"},{"word":"perfect matchings"},{"word":"combinatorial classes"},{"word":"pattern avoidance"},{"word":"combinatorial maps"},{"word":"triangulations"},{"word":"Catalan posets"},{"word":"uniquely sorted permutations"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/1qg5647z","frozenauthors":[{"first_name":"Lukas","middle_name":"","last_name":"Nabergall","name_suffix":"","institution":"Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada.","department":""}],"date_submitted":"2023-12-22T13:47:10Z","date_accepted":"2023-12-22T13:47:10Z","date_published":"2023-12-22T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64915/galley/49725/download/"}]}